Have multiple parts of model share same UV/texture
Mark Seams in edit mode and uv map as in image above.



Move UV island on top of the other as in image above. You can do a more precise job than I did.  I did not place them on exactly top of each other because I wanted to emphasize that both halves of the sphere have UV coordinates.  I wanted you too see both almost coincident.

If I did a perfect job and the islands were coincident that would be useful too.  So in the image/uv window you might think some pieces of the uv are missing when they are coincident.  That could be confusing.  Islands can be scaled small, merged to same point, or moved off the uv coordinate area.  So again they would be difficult to notice.

Select the vertices of the discussion, view othogonal, so that the vertices line up, then project.

Lastly there are three modifiers you might consider.  UV Warp, UV Project and Data Transfer.